OK, to touch on a few points here.
1. It was legal to build lowers (or even complete guns) during the AWB. The catch was, they had to be built as ban compliant. meaning no features that scared legislators that didn't know any better or did know better and just didnt' care. 2 or more of the following: Collapsible stocks, bayo lugs, pistol grip, flashider, threaded barrel, detachable mags, not more than 10 round mags, etc.
2. It is legal to transfer class 3 items to family members or anyone else, but it must be done through an SOT (special occupations tax holder) before they can take possession of it.
If you transfer a class 2 or 3 item before you're dead then the person buying it will have to pay the tax and fill out the forms and get an approval form from the DOJ/NFA/ATF and be transferred through the SOT dealer before they can actually have it. It's also usually a pay first thing, so money they pay for it (if any) will be tied up for quite some time.
After you're dead, the heir still has to apply for the licensing and fingerprints and background check through the FBI before they can "have" it, but is tax exempt in that case. There is a special form for those situations.
This is different from standard firearms, and didn't apply to the ban years either. This only applies to NFA firearms (including SBR's AOW's, Machine guns, Suppressors, Destructive Devices, etc)
3. When a firearm is transferred to you on a 4473, the serial number is logged on the form, but NICS does NOT!!!!!!!!!!!! ask for the serial number at any time, nor are they provided with it during a background check. Only after the dealer turns in his paperwork (goes out of business) is when the ATF is provided with the forms and the serial numbers, or during a routine audit.
This gets especially difficult on older firearms that do not have a serial number stamped on them at all. These were made prior to when it became law that it had to individually marked. Sorry, but I'm not sure what year that was off the top of my head, but firearms are floating around out there that are completely legal that does not have a serial number on them from the factory just because they're older than that legislation.
Tracing of a firearm serial number is usually initiated when a firearm is reported stolen.
The police agency reports this to the ATF (or maybe FBI) and the serial number is put in a database. Most of the time when a LEO asks for the serial number of your weapon they are only fishing to see if it's been stolen. Not if it's registered to you, or even who it's registered to unless of course you live somewhere that you must register them with the police.
The bad thing with the whole stolen serial number thing is, since the ATF doesn't ask for the serial when you actually buy a gun, it may already be registered as stolen before you buy it LEGALLY!!! This is something that I personally would like to see change, but that's not my call. In a situation like that, the dealer shows the ATF from the bound book where or who the firearm was aqcuired (bought from), and then the papertrail starts back to the other person or dealer that sold it to them, and so on. Though this doesn't help the person who has (er, had) the gun in the meantime.